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Let Pn denote the set of polynomials of the form ... with |a| ≤ 1 and |zk| ≥ 1 for 1 ≤ k ≤ n - m. For the polynomials of the form p(z) = z∏k=1n-1(z - zk), with |zk| ≥ 1, where 1 ≤ k ≤ n - 1, Brown [2] stated the problem "Find the best constant Cn such that p'(z) does not vanish in |z| < Cn". He also conjectured in the same paper that Cn = .... This problem was solved by Aziz and Zarger [1]. In this paper, we obtain the results which generalizes the results of Aziz and Zarger. [ABSTRACT FROM AUTHOR]

This paper considers the well-known Erdös–Lax and Turán-type inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. The obtained results produce inequalities that are sharper than the previous ones. Moreover, a numerical example is presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known. [ABSTRACT FROM AUTHOR]